Does k consistency always imply (k - 1) consistency?
Define P as a CSP where X, Y are the variables, domain of both is {1,2,3,4} and conditions in normal form are:


node-condition X&lt;4
arc-condition X=Y


P is 2-consistent (arc consistent) because for any X value it is possible to find a Y value that fulfills the arc-condition X=Y.

However, P is not 1-consistent (node-consistent) because exist a X value (X=4) that can not fulfill the node condition x&lt;4.

For these reasons, this problem is 2-consistent but not strongly 2-consistent.  

Obviously, it is straightforward to convert this example in a strongly 2-consistent problem, just reducing the domain to {1,2,3}.